Renouncing TPE simultaneous with signing a free agent?

[Garf]

Imagine a team needs to renounce a huge TPE in order to have enough cap space to sign a FA. My question:
Are those acts simultaneous, or do we check in between whether the renouncement took them so far below the cap that all other exceptions are lost too?

Concrete numbers:
1. Before: salaries & cap holds = 49.3M. TPE = 10M. MLE = 5.3M. BAE = 2.1M. In total, 66.7, thus no cap space.
2. After renouncement: salaries & cap holds = 49.3M. MLE = 5.3M. BAE = 2.1M. In total, 56.7, thus enough cap space to offer 6M+. But also momentarily so far below the cap that all other exceptions are lost if the renouncement and the signing are two separate acts.

[Garf]

Wait, I think I have just understood the idiocy of my question. If the team has cap space to sign the free agent, then also AFTER signing him it will be far enough below the cap to lose all other exceptions